7 TRACMASS—A Lagrangian Trajectory Model
247
proved TRACMASS. There are, however, still things that would be desirable to improve or add. The TRACMASS subgrid parameterizations, which were introduced
in Levine (2005), Döös and Engqvist (2007), Döös et al. (2011) will need to be
ameliorated with a higher order Markov model (see, e.g., Rupolo 2007).
It would also be desirable to evaluate the precision of the different TRACMASS
schemes in more detail and compare them with other trajectory schemes such as
the Runge–Kutta scheme. Fabbroni (2009) compared Ariane (Blanke and Raynaud
1997), which is based on the same equations as the time step version of TRACMASS with other trajectory schemes. The trajectories, simulated by Ariane, deviated clearly from the analytical solution and the other trajectory schemes in her
study and Ariane was concluded not to be as accurate as the other schemes. In the
present study we repeated the Fabbroni (2009) test of inertial oscillations, with exact
analytical solutions. We found in contrast to her test that the TRACMASS scheme
gave nearly exactly the same results as the analytical solution. The TRACMASS
time-stepping method, which is comparable to the Ariane method, requires, however, that one uses sufficiently intermediate velocity time steps between the GCM
time steps. The TRACMASS time-stepping method, when using 1000 intermediate
time steps, gave almost exactly the same precision as the TRACMASS the method
of analytical time integration. From these tests, we would like to argue that the
TRACMASS schemes give at least as accurate trajectories as any other scheme and
it is hard to argue that it would be of any use to have even more accurate schemes for
geophysical fluid applications given all the missing physics and scales in a GCM.
A more detailed and quantitive study would, however, be necessary to measure this.
One of the major advantages of TRACMASS is that it is mass conserving and
now can calculate all sorts of mass transports between different sections in the
ocean or the atmosphere as well as Lagrangian stream functions for chosen water/air masses. This can be particularly useful when performing analysis of, e.g.,
the inter-ocean exchange of water masses or the large scale atmospheric circulation.
TRACMASS has also turned out to be very useful in completely different applications such as studies of genetic connectivity, dispersion of radionuclides or identification of transport patterns in the surface layer as in the present book. The number
of possible TRACMASS applications will certainly continue to grow in the future
as long as GCMs are based on finite differences.
Acknowledgements The authors wish to thank Tarmo Soomere and Ewald Quak for constructive comments. This work was originally motivated by the BONUS+ project BalticWay that was
supported by the funding from the European Community’s Seventh Framework Programme (FP7
2007–2013) under grant agreement No. 217246 made with the joint Baltic Sea research and development programme BONUS+ and by the Swedish Research Council for Environment, Agricultural
Sciences and Spatial Planning (Formas, Ref. No. 2008–1900).
References
Berloff P, McWilliams J (2002) Material transport in oceanic gyres. Part II: Hierarchy of stochastic
models. J Phys Oceanogr 32:797–830
247
proved TRACMASS. There are, however, still things that would be desirable to improve or add. The TRACMASS subgrid parameterizations, which were introduced
in Levine (2005), Döös and Engqvist (2007), Döös et al. (2011) will need to be
ameliorated with a higher order Markov model (see, e.g., Rupolo 2007).
It would also be desirable to evaluate the precision of the different TRACMASS
schemes in more detail and compare them with other trajectory schemes such as
the Runge–Kutta scheme. Fabbroni (2009) compared Ariane (Blanke and Raynaud
1997), which is based on the same equations as the time step version of TRACMASS with other trajectory schemes. The trajectories, simulated by Ariane, deviated clearly from the analytical solution and the other trajectory schemes in her
study and Ariane was concluded not to be as accurate as the other schemes. In the
present study we repeated the Fabbroni (2009) test of inertial oscillations, with exact
analytical solutions. We found in contrast to her test that the TRACMASS scheme
gave nearly exactly the same results as the analytical solution. The TRACMASS
time-stepping method, which is comparable to the Ariane method, requires, however, that one uses sufficiently intermediate velocity time steps between the GCM
time steps. The TRACMASS time-stepping method, when using 1000 intermediate
time steps, gave almost exactly the same precision as the TRACMASS the method
of analytical time integration. From these tests, we would like to argue that the
TRACMASS schemes give at least as accurate trajectories as any other scheme and
it is hard to argue that it would be of any use to have even more accurate schemes for
geophysical fluid applications given all the missing physics and scales in a GCM.
A more detailed and quantitive study would, however, be necessary to measure this.
One of the major advantages of TRACMASS is that it is mass conserving and
now can calculate all sorts of mass transports between different sections in the
ocean or the atmosphere as well as Lagrangian stream functions for chosen water/air masses. This can be particularly useful when performing analysis of, e.g.,
the inter-ocean exchange of water masses or the large scale atmospheric circulation.
TRACMASS has also turned out to be very useful in completely different applications such as studies of genetic connectivity, dispersion of radionuclides or identification of transport patterns in the surface layer as in the present book. The number
of possible TRACMASS applications will certainly continue to grow in the future
as long as GCMs are based on finite differences.
Acknowledgements The authors wish to thank Tarmo Soomere and Ewald Quak for constructive comments. This work was originally motivated by the BONUS+ project BalticWay that was
supported by the funding from the European Community’s Seventh Framework Programme (FP7
2007–2013) under grant agreement No. 217246 made with the joint Baltic Sea research and development programme BONUS+ and by the Swedish Research Council for Environment, Agricultural
Sciences and Spatial Planning (Formas, Ref. No. 2008–1900).
References
Berloff P, McWilliams J (2002) Material transport in oceanic gyres. Part II: Hierarchy of stochastic
models. J Phys Oceanogr 32:797–830